(lol I started typing this back when Mario with Lasers posted)
Yeah, from what I observed using hacked Pokemon in Stadium 1/2 each Pokemon had a set of animations like Summon, Tackle, Punch, Shoot, etc that each move in the game was set up to use, as well as markers to where the Pokemon's mouth, feet, eyes, etc were, so the animations kinda took care of themselves from there.
Venusaur uses Ice Punch: Venusaur dives forward with icy mist on both front paws
Caterpie uses Fire Blast: It does the String Shot movements but instead shoots fire from its mouth
Magikarp uses Earthquake: It flops around a bit then BOOM when it smacks the ground from its highest jump
Pikachu uses Fly: It raises its arms and soars up, then plays its stand animation in midair, later playing its tackle animation instead of diving down
etc.
The problem is that there are double the Pokemon and moves now than there were back then, so it's more reasonable to stick with sprites.